Using a fast tree-searching algorithm and a Pentium cluster, we enumerated all the sequences and compact conformations (structures) for a protein folding model on a cubic lattice of size 4x3x3. We used two types of amino acids-hydrophobic (H) and polar (P)-to make up the sequences, so there were 2(36)approximate to6.87x10(10) different sequences. The total number of distinct structures was 84 731 192. We made use of a simple solvation model in which the energy of a sequence folded into a structure is minus the number of hydrophobic amino acids in the "core" of the structure. For every sequence, we found its ground state or ground states, i.e., the structure or structures for which its energy is lowest. About 0.3% of the sequences have a unique ground state. The number of structures that are unique ground states of at least one sequence is 2 662 050, about 3% of the total number of structures. However, these "designable" structures differ drastically in their designability, defined as the number of sequences whose unique ground state is that structure. To understand this variation in designability, we studied the distribution of structures in a high dimensional space in which each structure is represented by a string of 1's and 0's, denoting core and surface sites, respectively.